Some Aspects of Gauge Theories on Noncommutative Spacetime Xavier Calmet University of Brussels (ULB)

Outline Motivations and goals Local gauge symmetries on noncommutative spaces Space-Time symmetries of noncommutative spaces Gravity on noncommutative spaces Conclusions

Motivations Space-time noncommutativity is an extension of quantum mechanics: Heisenberg algebra: is extended with new noncommutative (NC) relations: that lead to new uncertainty relations:

This is a nice analogy to the Heisenberg uncertainty relations. Quantum mechanics and general relativity considered together imply the existence of a minimal length in Nature: Gauge theories with a fundamental length are thus very interesting. A class of models with a fundamental length are gauge theories on noncommutative spaces (length   ). Noncommutative coordinates appear in nature: e.g. hydrogen atom in a strong B field (first Landau level can be described in terms of NC coordinates). Tools which are developed can prove useful for solid states physics.

Idea of a noncommutative space-time is not new! It can be traced back to Snyder, Heisenberg, Pauli etc. At that time the motivation was that a cutoff could provide a solution to the infinities appearing in quantum field theory. Nowadays, we know that renormalization does the job for infinites of the Standard Model, but modifying space-time at short distances will help for quantum gravity.

Furthermore, the Standard Model needs to be extended if it is coupled to gravity since it is then inconsistent: noncommutative gauge theories are a natural candidate to solve this problem. Another motivation is string theory where these noncommutative relations appear. But the situation is in that case very different!

Gauge Symmetries on Noncommutative Spaces

Goals How does the Standard Model of particle physics which is a gauge theory based on the group SU(3)  SU(2)  U(1), emerge as a low energy action of a noncommutative gauge theory? The main difficulty is to implement symmetries on NC spaces. We need to understand how to implement SU(N) gauge symmetries on NC spaces. Are there space-time symmetries (Lorentz invariance) for noncommutative spaces?

Symmetries and Particle Physics commutative space-time case Impose invariance of the action under certain transformations. Two symmetries are crucial in order to formulate the Standard Model of particle physics: -Space-time: Lorentz invariance, and combinations of C, P and T e.g.: -Local gauge symmetries

Enveloping algebra approach to NC Goal: derive low energy effective actions for NC actions which are too difficult to handle. Strategy: map NC actions to an effective action on a commutative space-time such that higher order operators describe this special property of space-time. There is an alternative to taking fields in the Lie algebra: consider fields in the enveloping algebra

Definitions and Gauge Transf. def. 1: consider the algebra algebra of noncommutative functions def. 2: generators of the algebra: ``coordinates´´ def. 3: : elements of the algebra infinitesimal gauge transformation: note that the coordinates do not transform under a gauge transformation:

one has: that’s not covariant! Introduce a covariant coordinate such that i.e. let’s set this implies: that’s the central result: relation between coord. gauge fields and Yang-Mills fields! That’s not trivial: problem with direct product!

Star product & Weyl quantization def: commutative algebra of functions: aim: construct a vector space isomorphism W. Choose a way to “decompose” elements of : (basis):

we need to def. the product (noncommutative multiplication) in : Weyl quantization procedure: Let us use the Campbell-Baker-Hausdorff formula: We then have:

to leading order: We now have the first map: we know how to replace the argument of the functions, i.e. the NC coordinates by usual coordinates: price to pay is the star product. This is done using the isomorphism. The second map will map the function, this second map (Seiberg-Witten map) is linked to gauge invariance, more later.

Field Theory Let us start from the relations: the Yang-Mills gauge potential is defined as has the usual transformation property: The covariant coordinate leads to the Yang-Mills potential!

Local gauge theories on NC spaces Let be Lie-algebra valued gauge transformations, the commutator: is a gauge transformation only for U(N) gauge transformations in the (anti)fundamental or adjoint representation. Problem: Standard Model requires SU(N)! BUT, it can close for all groups if we take the fields and gauge transformations to be in the enveloping algebra: Is there an infinite number of degrees of freedom? No! They can be reduced using Seiberg-Witten maps!

Consistency condition and Seiberg-Witten map 1.Replace the noncommutative variable by a commutative one. Price to pay is the introduction of the star product: 2.Let us consider the commutator once again:

Let us now assume that are in the enveloping algebra: one finds in 0th order in  and in the leading order in .

Previous partial differential equation is solved by: Expanding the star product and the fields via the SW maps in the leading order in theta, one finds:

SM on NC Space-Time Problems: a) direct product of groups b) charge quantization c) Yukawa couplings d) “Trace” in the enveloping algebra Solutions: a)One can’t introduce 3 NC gauge potentials: must remain covariant! solution: introduce a master field: SW map for Note that

b) solution to charge quantization: introduce n NC photons: Too many degrees of freedom? No  Seiberg-Witten map! there is only one classical photon!

c) Yukawa couplings: left/right makes a difference!  Complication for Yukawa couplings: is not NC gauge invariant if transforms only on the r.h.s. or l.h.s. Solution: Hybrid SW map: with note: it is possible to couple neutral particles to the photon in a gauge invariant way!

d) trace for the gauge part of the action: is a huge matrix. There is not a unique way to fix the trace, gauge inv. only requires: Minimal model:

Other choice

Bounds on NC scale From colliders: Lots of corrections to SM processes, but large background: search for rare decays. Smoking gun for NC: Z-->   or Z--> g g. Limit on  NC from LEP is below 1 TeV. From low energy experiments: Bounds on imply  NC  TeV from atomic clock comparison (Be 9 ). Note that the bound comes from Lorentz violation, and is thus not a “direct” test of the noncommutative nature of space-time.

Quantum Mechanics and EDM There are claims in the literature that EDMs can put very tight bounds on the scale for spacetime noncommutativity. A formulation of Quantum Mechanics on a NC spacetime is needed to address this question. Let us start from the QED action on a NC spacetime:

Two maps lead to the following action:

And the Dirac eq. easily follows: Let us now prepare the non-relativistic expansion: And we

From this it is easy to obtain the low energy Hamiltonian: 3 operators are CP violating:

Let us look at one of them: However it is not of the shape: Experiments searching for an EDM are not sensitive to this operator: there is no bound! These experiments measure the energy difference between a two-levels system. Here the effect cancels out.

Space-time symmetries of NC spaces Consider NC: Furthermore, one has the Heisenberg algebra: Let us now do a variable transformation: It leads to the following algebra:

Let us consider transformations of the commuting coordinates: one also has The invariant length is given by: It is invariant if We can now implement this transformation for the NC coordinates:

The invariant length is given by: the derivative is given by: it transforms as under a noncommutative Lorentz transformation.

The NC Yang-Mills potential transforms as: and the covariant derivative as: The field strength transforms as: and a spinor as:

This represents an extension of special relativity. The limit   0 is well defined: one recovers the usual Lorentz invariance. Note: we do not deform the Poincaré algebra! It is easy to verify that the actions discussed previously are indeed invariant under these transformations. This symmetry is important because bounds on space-time noncommutativity come from bounds on Lorentz violation (atomic clocks). The bounds will be affected. Any operator derived from loop calculations must be invariant under this symmetry: beware of artifacts of regularization procedure.

Let us look at the light cone of a photon on a NC spacetime: which is not ! Let us now compute (at equal time) the expectation value of the commutator between and as done by Greenberg. It is proportional to He concludes that microcausality is violated. However this precisely corresponds to our light cone: microcausality is not violated! Is microcausality violated?

Quantization of Noncommutative QED Misuse of the term effective theory: mapped theory? Seiberg-Witten expansion is an expansion in. If one expands in and then quantize the theory (expansion in terms of one can miss important resummation effects. This is indeed the case because of the vertices phases as we shall see. Let us start from the unexpanded action: Fields are representation of the Lorentz group: quantize the fields which are in the enveloping algebra.

Add Faddeev-Popov terms Feynman rules are then given by:

It is then straightforward to compute the beta function And the renormalized vertex : The quantized and renormalized action can then be mapped on a commutative spacetime. The vertex correction is given by

Gravity on Noncommutative Spaces

Gravity on noncommutative spaces Hypothesis:  is a constant of nature and it has the same value in every coordinate frame. Well if that is the situation, what are the coordinate transformations allowed by the NC algebra: Let us consider the transformations: and study the NC algebra: It is invariant iff The solutions are: They form a subgroup of 4-Volume preserving coord. transformations.

We now want to implement this symmetry for a NC gravity action. We consider iso(3,1) but restrict ourselves to the coord. transformations that preserve  Consider the enveloping algebra: Consistency condition:

Differential equations: 0th order in  : 1st order in  : Solution: Now for the spin connection: One thus has:

For the field strength one has: classically one has: Note that our covariant derivative is torsion free. Field strength for the local Lorentz symmetry: Noncommutative Riemann tensor:

we can then define a noncommutative Ricci tensor: and a noncommutative Ricci scalar: It is easy to see that the leading order correction is vanishing: with which vanishes! Second order corrections have to be calculated. However the result is quite complicated. It implies a that we need to know solutions to the consistency conditions to second order in theta. OK get ready for the result:

Action: Equations of motion Can be massaged into: Remarkable: on a canonical NC spacetime: the cosmological constant is an integration constant uncorrelated to parameters of the action!

Some open questions Understand how to formulate gravity on NC spaces (non constant  ). CMB signatures of NC physics? Do NC black holes have singularities? What about horizon? Hawking radiation? Information loss? Is quantum gravity formulated on NC spaces renormalizable? Fine tuning issues and NC physics. Certain short distance modifications of space-time can modify the high energy behavior of loops. New ideas to break gauge symmetries: after all lots of ideas come from solid state physics and we have quite a few models in solid state physics that are described by NC gauge theories. This will lead to new phenomenology for the LHC.

Conclusions Noncommutative gauge theories are examples of non-local theories with a minimal length. SU(N) gauge symmetries, which are crucial for the Standard Model, can be implemented on noncommutative spaces. Noncommutative Lorentz transformations can be introduced. Constraints are not very severe! Bounds of the order of few TeV only. Lots of open issues e.g. Loops, general relativity: work in progress! Applications to solid state physics, cosmology etc. Even if there is a grand desert, Planck scale physics might be accessible with low energy experiments. Exciting field in development.